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Two-dimensional Vorticity Dynamics
Two-dimensional Vorticity Dynamics
u =
u(x, y, t)v(x, y, t)
0
ω =
00
ω(x, y, t)
ω =∂v
∂x− ∂u
∂y
Consider the stream function ψ(x, y)
A = ψ(x, y)ez u = ∇× A
ω = −∆ψez
ψ(x, y, t) = − 12 π
ω(x, y, t) ln(R) dx dy + ψp
R2 = (r − r)2 = (x− x)2 + (y − y)2
u =1
2 π
ω(x, y, t)ez × (r − r)
|r − r|2 dx dy +∇φ
Biot and Savart Law: 2D Flows
For large r it decreases algebraically Γ/(2πr)
Dω
Dt= ν∆ω
Cancellation of Vorticity
Diffusion of Vorticity
ν = 0Viscous Flow
∂ω
∂t= ν
∂2ω
∂y2
Vorticity diffuses by Viscous Effect
A unique vortex layer
ω ∼ Γδ(t)
ω =Γ√4νt
exp(− y2
4νt)
Circulation is conserved
t = 0
ω = ω0 sin(ky)An array of vortex layers
∂ω
∂t= ν
∂2ω
∂y2
ω(y, t) = ω0 exp(−k2t) sin(ky)
Vorticity Cancellation by Viscous Effect
a2 ≡
ω | CM |2 dS ω dS
Γ =
ω dS
C : Position of the vorticity centroid
Size of total vorticity repartition
a2 = a20 + 4 ν t
OC =
ω OM dS ω dS
The total vorticity field ν = 0
a2 ≡
ω | CM |2 dS ω dS
Γ =
ω dS
C : Position of the vortex center the centroid
Size of the vortex
a2 = a20 + 4 ν t
OC =
ω OM dS ω dS
Unique Vortex ν = 0
Conservation of vorticity pointwise
ν = 0
Size of any given vortex (only for a uniform vorticity)
Conservation of circulation of any given vortex
Inviscid vortex flow
⇒
⇒
In an assembly of vortices this means :
|r| << |r|
ln(R) =12
ln |r − r|2 =
12
lnr2
1−
2r2
r · r
∼ ln(r)−r · r
r2+ O
a2
L2
ψ = −ln(r)2 π
ω dx dy
Γ
+r
2 π r2
r ω dx dy
rC=0
+O
a2
r2
OC =
ω OM dS ω dS
Potential vortex flow
ψ ∼ −Γ2 π
ln(r) point vortex
+O
a2
r2
At a point far away from a vortex location, the velocity field produced by a vortex is similar to a
point-vortex velocity ω = Γδ(x− xc)δ(y − yc)
ur(r) =1r
∂ψ
∂θ= 0, uθ(r) = −∂ψ
∂r=
Γ2 π r
Potential vortex flow
For the velocity field at M, only two quantities matter
The distance OM to the vortex centroid O
The vortex circulation Γ
Γj
rj(xj , yj)
r(x, y)
M
uVelocity due to vortex j at point r(x, y)
uj =Γ
2 π |r − rj |2
−(y − yj)
x− xj
0
xi
yi
=
j =i
Γj
2 π R2ij
−(yi − yj)
xi − xj
, R2
ij = |ri − rj |2
A system of point vortices : Dynamical system
H = − 14 π
N
k=1
j =k
Γk Γj ln(Rkj)
This system is an Hamiltonian system
pi ≡ Γi yiqi ≡ xi
qi =∂H
∂pi, pi = −∂H
∂qi
H is a conserved energy
Interaction energy
Kinetic energy is infinite
xC
yC
=
1Γ
N
i=1
Γi
xi
yi
A centroid
A dispersion radius
d2 =1Γ
N
i=1
Γi [(xi − xC)2 + (yi − yC)2]
Other Conserved Quantities
System of Two point vortices
r1
r2Γ1
Γ2
Γ1 x1 + Γ2 x2 = cteΓ1 y1 + Γ2 y2 = cte
3 Conserved quantities
4 dynamical variables x1, y1, x2, y2
The dispersion radius gives again R12 = |r1 − r2| = Cst
H = − 12 π
Γ1 Γ2 ln(R12) = Cst ⇒ R12 = |r1 − r2| = Cst
Two point vortices Γ1Γ2 > 0
C is located between 1 and 2.
The system rotates around C at angular velocity
Ω =Γ1
2 π R12
Γ1 + Γ2
Γ1 R12=
Γ1 + Γ2
2 π R212
Two point vortices Γ1Γ2 < 0
C is located on the side of the more intense vortex
Two Point Vortices: a Dipole Γ1 + Γ2 = 0
The dipole does not rotate, and translates at velocity
Γ/(2π R12)streamlines in the comoving Frame
(a) (b)
streamlines in the Laboratory
Trapping and Transporting Fluid
Three Point Vortices
Simple dynamics : an integrable motion
Triangle shape oscillates with a period T1
Triangle rotates with a period T2
T1
T2is rational number (initial condition) ⇒ periodic
T1
T2
is in general not in an irrational number
⇒ Quasi-periodic Motion
Four or more point Vortices
The dynamical system is non integrable: chaotic
4 Conserved quantities
8 dynamical variables
Point Vortices with a Wall
(a) (b)
A unique vortex along a wall A Dipole along a wall
Image theory
Infinite Row of Point Vortices
Pairing instability
From an Infinite Row of Point Vorticestowards
Vortex Sheet
Vortex Sheet
self-similar spiralling
Roll-up behind a wing
Vortex sheet near trailing edges of a Delta wing
A First Vortex sheet with zero total circulation
Another Vortex sheet with non nul vortex circulation
EXTENDED 2D VORTICES
Vortex axisymmetrization
Vorticity Streamfunction
Stripping
Vortex in a Potential Strain field
Vortex along a wall or a Dipole
Gerris (S.Popinet)
Two-dimensional Turbulence : Inverse Cascade
Merging of Vortices
Vortex Merging
Josserand and Rossi 2005
Vortex Dipole
Vortex In a bath
ω(y, z, t = 0) = ω(+)(y, z) + ω(−)(y, z)
ω(±)(y, z) = ± Γ0
πa20
exp− (y ∓ b0/2)2 + z2
a20
!0
a0
b0y
z
0 200 400 600 800 10000
0.2
0.4
0.6
0.8
1
1.2
t
!
I
!h
Cancellation of Vorticity
t = 0 20 40
50 90 1000
1 0.5 0 0.5 11
0.5
0
0.5
1
1.5
2
2.5
3t = 0 t = 8
Strain dipole in the Plane
Numerical Simulation Shallow Water equations (Code Gerris S.Popinet)